Verification transcript for Cairn Commons claim C-8: C(61,25,3) <= 27
Author: Caleb Ashton

== The 27 blocks (contents of 61.25.3.txt), one 25-subset of {1..61} per line ==
5 6 8 11 13 16 17 23 26 27 28 30 31 33 41 44 45 46 48 49 51 53 55 58 61
2 4 5 6 7 11 13 17 20 24 30 31 32 34 40 45 46 47 50 52 56 58 59 60 61
1 5 6 10 11 12 13 15 17 18 19 22 29 30 31 35 38 39 42 45 46 54 56 58 61
1 3 9 13 14 16 18 20 27 28 29 30 31 32 33 34 38 40 41 42 46 50 51 52 53
1 3 4 5 6 10 14 17 18 20 21 22 23 32 34 37 38 39 40 42 43 50 54 57 61
1 5 6 10 17 18 21 23 26 28 29 32 33 34 37 38 40 41 50 53 54 55 56 59 61
4 8 9 10 11 22 28 32 33 34 39 40 41 43 44 45 48 49 50 52 53 54 56 57 58
2 3 5 6 7 9 11 13 14 17 21 24 25 30 31 36 37 43 45 46 47 56 57 58 61
2 3 4 7 10 11 14 16 22 23 24 25 26 27 29 36 39 45 47 51 52 54 55 56 58
2 3 4 5 6 7 12 14 15 17 19 20 23 24 26 28 33 35 41 43 47 53 55 57 61
2 7 9 10 16 20 21 22 24 26 27 29 32 34 37 39 40 43 47 50 51 54 57 59 60
8 11 13 16 21 23 26 27 29 30 31 32 34 37 40 44 45 46 48 49 50 51 55 56 58
2 7 10 12 13 15 16 19 22 24 27 28 29 30 31 33 35 39 41 46 47 51 53 54 56
10 12 13 15 19 20 21 22 25 30 31 32 34 35 36 37 39 40 43 46 50 52 54 55 57
1 3 4 11 12 14 15 18 19 21 25 26 32 34 35 36 37 38 40 42 45 50 52 56 58
3 5 6 8 12 14 15 16 17 19 21 27 29 35 37 43 44 48 49 51 52 57 59 60 61
1 4 13 16 18 21 25 27 28 29 30 31 33 36 37 38 41 42 43 46 51 53 57 59 60
2 7 9 12 15 19 21 23 24 25 28 32 33 34 35 36 37 40 41 47 50 52 53 59 60
1 2 7 8 11 12 15 18 19 24 28 29 33 35 38 41 42 44 45 47 48 49 53 56 58
1 5 6 9 10 17 18 22 23 25 26 28 33 36 38 39 41 42 52 53 54 55 59 60 61
4 5 6 8 9 12 15 16 17 19 20 25 27 29 32 34 35 36 40 44 48 49 50 51 61
1 2 7 8 13 18 20 23 24 25 26 30 31 36 38 42 43 44 46 47 48 49 52 55 57
1 2 5 6 7 8 10 16 17 18 22 24 27 29 38 39 42 44 47 48 49 51 54 56 61
1 9 11 12 15 16 18 19 20 23 26 27 29 35 38 42 43 45 51 55 56 57 58 59 60
3 8 10 11 14 20 21 22 25 28 33 36 37 39 41 44 45 48 49 53 54 56 58 59 60
1 2 3 4 7 8 9 14 18 21 24 32 34 37 38 40 42 44 47 48 49 50 55 59 60
3 4 8 9 10 12 13 14 15 19 22 23 26 30 31 35 39 44 46 48 49 54 55 59 60

== Verification script (verify.py) ==
import itertools, math, hashlib
data = open('61.25.3.txt','rb').read()
print("SHA-256:", hashlib.sha256(data).hexdigest())
B = [list(map(int, l.split())) for l in data.decode().splitlines() if l.strip()]
print("blocks:", len(B))
print("all blocks size 25 with distinct elements:", all(len(b)==25 and len(set(b))==25 for b in B))
print("all points in 1..61:", all(1<=x<=61 for b in B for x in b))
cov = {t for b in B for t in itertools.combinations(sorted(b), 3)}
print("triples covered:", len(cov), "of", math.comb(61,3))
print("uncovered:", math.comb(61,3)-len(cov))

== Output of: python3 verify.py ==
SHA-256: 097a9067d0628e1d3ba01f6230bafbcab3573b78f4d0c9913b8759676e00a18e
blocks: 27
all blocks size 25 with distinct elements: True
all points in 1..61: True
triples covered: 35990 of 35990
uncovered: 0

Prior best 28: https://ljcr.dmgordon.org/cover/show_cover.php?v=61&k=25&t=3